Inertial Mass as Speedometer: Physics & Universe

[metiman]

TL;DR Summary

The idea of relativistic or inertial mass or that accelerating a given mass requires more and more force as velocity increases led me to wonder if inertial mass measurement could be used to detect speed from inside a sealed box.

I get that the concept of relativistic mass has sort of been deprecated in physics these days and that relativistic momentum is supposed to be seen as more well useful. So let momentum equal $\mathtt ~~ \frac {mv} {\sqrt {1 - \frac {v^2} {c^2}}} ~~$ or $~~{mv\gamma}~~$. So mass is supposed to be invariant even though the force required to accelerate that mass keeps going up as v increases. Fair enough but could this increase in the force or energy required to accelerate a given mass at relativistic speeds be used as a speed detector? Let's assume the ship has reached 0.75 c and has cut the engines and is coasting so as to avoid general relativity if possible. Could you just weigh a 1.000000 kilo piece of aluminum when you start your trip and weigh it again after you accelerate to say .75c to determine your exact speed relative to... something? I am not clear on what reference frame you would be measuring against. I suppose it would relative to the start of the trip.

Is there some reason why this change in effective mass or inertial mass or relativistic mass or whatever you want to call the change in force required to accelerate a reference object could not be used to measure your speed?

I am also wondering why you cannot somehow 'reset' your frame of reference to that of the ship itself now traveling at 0.75c so that you can keep accelerating at the same rate as you did at the start of your trip given the same constant force propulsion system. It is like the universe knows your history. It knows that you have already accelerated to a high speed with respect to something. It knows your kinetic energy is insanely high.

If there really are no privileged frames of reference this seems like an odd result. After all the Earth itself is traveling very fast around the sun and the sun is whizzing around the center of the galaxy also at great speed and the whole galaxy itself is not fixed either. Although I suppose none of these are relativistic speeds and so can probably be discounted, but what if we imagine that the sun was traveling at relativistic speed with respect to some reference. Is all of that motion taken into account when the universe decides you are going too fast and reduces your acceleration according to $\gamma$ ? Is there no way to trick the universe into thinking you were always traveling at that speed? Can you hack the universe's acceleration log or journal to at least reach c more easily? Do you see the trouble I am having with all of this? How can the universe possibly know how fast you are going and limit your acceleration accordingly if there is no privileged frame of reference?

I was also wondering if an astronaut inside a relativistic ship would become so heavy (as in massive) that they would no longer be strong enough to move themselves. I imagine arm and leg muscles far too weak to push their own 5000 kg body for instance. Would they be fixed in the same location on the ship forever unless the ship accelerated around them?

 

[Ibix]

Could you just weigh a 1.000000 kilo piece of aluminum when you start your trip and weigh it again after you accelerate to say .75c to determine your exact speed relative to... something?

By "weigh" I take it you mean to apply a little kick to it and see how hard it is to accelerate? You can do that. The relevant velocity and gamma factor are the ones between the mass and your measurement devices which, if they are sealed in the box along with the mass, are $v=0, \gamma=1$.

It's important to remember that the velocity you use in any calculation does not have a universally agreed value. You can only measure velocity relative to something. Typically, that something will be the device that does the measurement - which gets you zero velocity in your example.

 

[metiman]

Well the sealed box is a ship that has accelerated to 0.75c and is now coasting with the engines off. If the velocity is zero then my constant force engine can accelerate the ship at the same rate it did when the trip started, right? Wouldn't it mean the ship would have zero momentum and zero kinetic energy?

It sounds like you are saying this inertial mass based speedometer would always just read zero and so would not be particularly useful. Can you explain why? I thought it would take more force to 'kick' 1 kg at a speed of 0.75c than at a speed of zero. It sounds like there is some error in my thinking with respect to reference frames and changes in relativistic momentum.

 

[Ibix]

You edited while I was replying

I was also wondering if an astronaut inside a relativistic ship would become so heavy (as in massive) that they would no longer be strong enough to move themselves.

No. As above, the astronaut can regard himself as at rest, so sees nothing unusual and can move freely. Analysis from any other frame may be more complicated, but will lead to the same conclusion.

Relativity is based on the principle of relativity. Its description of any "sealed box" experiment must be consistent with the box not moving, however anyone chooses to regard it as moving.

 

[Ibix]

If the velocity is zero then my constant force engine can accelerate the ship at the same rate it did when the trip started, right? Wouldn't it mean the ship would have zero momentum and zero kinetic energy?

As measured by devices on board the ship, yes. The ship is at rest with respect to devices attached to it.

It sounds like you are saying this inertial mass based speedometer would always just read zero and so would not be particularly useful. Can you explain why?

It follows from the principle of relativity. The speedometer may always regard itself as at rest and hence, in your setup, the mass as at rest. So the result will always be the same.

I thought it would take more force to 'kick' 1 kg at a speed of 0.75c than at a speed of zero.

If the 1kg mass is doing 0.75c relative to whatever is providing the kick, yes it will. But the mass is at rest with respect to your speedometer, not doing 0.75c. So you will measure speed zero.

 

[Orodruin]

Also note that the relationship between acceleration and force is not a simple scalar in relativity for objects that are in motion.

Energy is also not independent of the inertial system, nor is momentum. This is true already in classical mechanics.

How can the universe possibly know how fast you are going and limit your acceleration accordingly if there is no privileged frame of reference?

It cannot. This is the entire point of relativity (special as well as Galilean) as the basic assumpion is that there is no privileged frame. Therefore anything that is derived from this assumption must be compatible with it. Anything that seems to be contradicting it is either false or not correctly interpreted (or both).

 

[Orodruin]

It sounds like you are saying this inertial mass based speedometer would always just read zero and so would not be particularly useful.

It would be useless by definition since there is no universal speed to be measured. Any number it would show would be nonsense.

I thought it would take more force to 'kick' 1 kg at a speed of 0.75c than at a speed of zero. It sounds like there is some error in my thinking with respect to reference frames and changes in relativistic momentum.

Force is not a quantity that is independent of the inertial frame (much like energy and momentum). So yes, in a frame where the object travels at 0.75c you would require more force to achieve the same acceleration. However, all of the concepts involved here are dependent on the inertial frame. The acceleration, the force, the momentum, and the inertia. All depend on the inertial frame (and direction of motion/acceleration) in a non-trivial manner. However, in the instantaneous rest frame, the inertia is always m in all directions.

 

[metiman]

How enjoyably strange. But the ship really will be harder to accelerate after getting to a high relativistic speed, right? For a more concrete example let's assume a 10 million kg ship with an improbability drive that can accelerate it at exactly 1 G at the start of the trip when the motion of the ship is similar to that of the earth-moon-sun system. It is launched from a lunar base toward K2-18 124 light years away with this miracle drive. However the miracle drive can only produce a constant thrust of however many Newtons are required to accelerate the ship at 1G from rest. It cannot keep increasing the thrust as the apparent / inertial / relativistic mass of the ship increases.

The thrust or force required from this engine (to maintain a constant rate of acceleration) at high relativistic speeds like 0.75c will be vastly greater than at the start of the trip, correct? Even though the engine is at the same reference frame as the astronaut. It seems like there may not even be a way to measure the incredible speed achieved with this miracle space drive. And yet somehow a great deal more energy is required to get closer to c.

 

[Orodruin]

But the ship really will be harder to accelerate after getting to a high relativistic speed, right?

In the inertial frame where it has that high relativistic speed, yes. However, there is no way you can unambiguously claim that an object has a high relativistic speed without reference to a particular inertial frame.

However the miracle drive can only produce a constant thrust of however many Newtons are required to accelerate the ship at 1G from rest. It cannot keep increasing the thrust as the apparent / inertial / relativistic mass of the ship increases.

The thrust or force required from this engine at high relativistic speeds like 0.75c will be vastly greater than at the start of the trip, correct? Even though the engine is at the same reference frame as the astronaut. It seems like there may not even be a way to measure the incredible speed achieved with this miracle space drive. And yet somehow a great deal more energy is required to get closer to c.

In the original reference frame (the Earth frame), the acceleration will decrease (the force in the Earth frame will also change though). However, the acceleration will always be 1g in the ship's instantaneous rest frame.

No, you cannot measure an absolute speed. You can only measure relative speeds. Again, energy is not something absolute either. It is also dependent on the inertial frame.

 

[Ibix]

But the ship really will be harder to accelerate after getting to a high relativistic speed, right?

Using the word "really" is a mistake. It suggests you are thinking of this as some kind of confusing trick. It's much better to think of measurements, because the point is that although measurements may differ they are all equally valid. And one consistent set of measurements can be used to deduce someone else's measurements.

If your ship is supposed to be able to provide 1g of acceleration as measured by accelerometers on the ship then it will always be able to. Observers at rest in an inertial frame will measure its acceleration to be decreasing as it accelerates, though. Such an observer could transform their measurements into the ship's frame and deduce that shipboard observers were measuring 1g.

 

[metiman]

So it sounds like I should be thinking in terms of three reference frames. One based on the launch location and another on the motion of the ship after it has accelerated to whatever relativistic speed I cut thrust at so as to have another inertial frame. And another at the destination of this one way trip 124 light years away.

From the pov of observers on Earth everything would happen as I first assumed. The ship's acceleration would get slower and slower until it was basically stuck at a certain speed relative to the earth. From the pov of the ship and astronauts they would continue to experience an artificial gravity of 1g from the direction of the engine thrust even after 0.75c however their speed relative to both the Earth and K2-18 would seem to stop increasing at some point despite the fact that they continue to feel 1g? So it's a bit like some magical force, the hand of a god, is holding them back from increasing their speed despite the 1g of acceleration.

It would actually take about 93 years (neglecting any effects from time dilation) from the pov of the astronauts even if from their pov they have been constantly accelerating the whole trip at 1g and should have reached c within the first year of the trip. I am thinking about what 'bothers' me about this whole thing and I guess it is that it still seems like the universe somehow knows that the spaceship is getting close to the speed of light because it is able to slow down and stop the acceleration of the craft at some point.

Even if it doesn't seem like that is happening from within the spaceship it will still take the amount of time to travel the distance consistent with being frozen at say 0.75c. Since nothing can travel faster than c or really even be allowed to reach it it would make sense that the universe can somehow sense when anything is getting close to that speed. I suppose that is the part I find most confusing. How does it measure your speed to apply the Lorentz factor and reduce your acceleration? It seems to be able to do this from any reference frame and apply some push back.

 

[A.T.]

I guess it is that it still seems like the universe somehow knows ...

Yeah, the universe keeps track of everything that happens and applies it's laws, like some control freak.

 

[metiman]

Well it's laws can get awfully strange. This slowing down of acceleration in high speed reference frames seems like a kind of asymmetry. It seems like if you could somehow measure this reduction in acceleration from within the spaceship it would be a way to tell your new 0.75c inertial reference frame is different from the one you started at. What if you looked out a window and noted the color of the starlight was doppler shifted? Could you measure your speed that way?

Are we totally sure this is not an exception to the no privileged referenced frame thing? It's not the slowing down itself that bothers me. It's the apparent violation of the all inertial reference frames are equally valid rule. The increase in (unmeasurable from the local frame) apparent mass only happens when the reference frame is traveling at close to c with respect to something. Possibly with respect to anything in the universe.

Does anyone know of any books that go into depth exploring this particular aspect of SR?

 

[Ibix]

So it sounds like I should be thinking in terms of three reference frames. One based on the launch location and another on the motion of the ship after it has accelerated to whatever relativistic speed I cut thrust at so as to have another inertial frame. And another at the destination of this one way trip 124 light years away.

Assuming the destination star is at rest with respect to the start star, these two are the same. More generally, you can work in any reference frame you like, although the maths simplifies in some. For example, if you want to understand a particular person's point of view and determine what their measurements would be, their rest frame is the easiest one to pick.

From the pov of observers on Earth everything would happen as I first assumed. The ship's acceleration would get slower and slower until it was basically stuck at a certain speed relative to the earth.

They will always continue to accelerate as long as their engine is on. The rate as measured in this frame may drop immeasurably low, though, if that's what you are trying to say.

From the pov of the ship and astronauts they would continue to experience an artificial gravity of 1g from the direction of the engine thrust even after 0.75c however their speed relative to both the Earth and K2-18 would seem to stop increasing at some point despite the fact that they continue to feel 1g?

How the astronauts interpret their measurements while under acceleration is a rather complicated topic. However, if they were to accelerate in short bursts and wait long enough to be able to use an inertial frame covering the stars again, they would find each burst results in a smaller and smaller increase in the speed of the stars relative to them. Again, though, the speed would never stop increasing as long as they kept accelerating, although it would eventually become immeasurably small as the speed approaches that of light.

So it's a bit like some magical force, the hand of a god, is holding them back from increasing their speed despite the 1g of acceleration.

No. They will feel no forces except those they exert through their rockets. There is no magic in this.

It would actually take about 93 years (neglecting any effects from time dilation)

You said the star was 124 light years away. If you are neglecting the effects of time dilation (a silly thing to do, since you are accelerating to relativistic speed) it cannot possibly take less than 124 years without exceeding the speed of light.

I am thinking about what 'bothers' me about this whole thing and I guess it is that it still seems like the universe somehow knows that the spaceship is getting close to the speed of light because it is able to slow down and stop the acceleration of the craft at some point.

The point is that the ship can always regard itself as at rest. From its own point of view, it always has to accelerate another $3\times 10^8m/s$ to reach light speed, so it is always free to accelerate as hard as it likes. Other people's descriptions of this process will be different, but it isn't anything to do with anything "knowing" you are approaching the speed of light. Quite apart from the unnecessary anthropomorphism, that would imply that you are approaching the speed of light in some absolute sense. You aren't. You can always regard yourself as at rest.

Even if it doesn't seem like that is happening from within the spaceship it will still take the amount of time to travel the distance consistent with being frozen at say 0.75c.

As I say, handling acceleration is complicated - you need to introduce calculus and some discussion of synchronisation conventions. However, if we assume that the ship accelerates instantaneously to 0.8c (the numbers are easier to work with at that speed than 0.75c) then the Earth frame says that the ship takes 124/0.8=155 years to reach the star. It has a $\gamma$ of 5/3, so the shipboard clocks read only 3/5 of that, or 93 years.

In the ship frame, however, the star starts 74.4 light years away, due to the effects of length contraction and the relativity of simultaneity, and takes 74.4/0.8=93 years to reach the ship. It's all consistent.

How does it measure your speed to apply the Lorentz factor and reduce your acceleration? It seems to be able to do this from any reference frame and apply some push back.

This makes no sense. I think you really need to do three things. The first is to forget about acceleration for the time being. It introduces a lot of complexity that you don't seem ready to handle (that's not an insult - there is an awful lot to unlearn and more to learn before relativity fits together, and inertial motion is enough to begin with). The second is to look up the Lorentz transforms, which relate the position and time of events measured by one frame to those measured by another. They are the basis of everything in relativity, and their implications (especially the relativity of simultaneity) are very important. Thirdly, you need a proper textbook if you want to learn. Taylor and Wheeler's Spacetime Physics is good, and the first chapter is freely available online if you want to take a look. If you don't want to buy a physical book, a former Mentor here, Ben Crowell, wrote a textbook that is free to download in its entirety from www.lightandmatter.com/books.

 

[Ibix]

It seems like if you could somehow measure this reduction in acceleration from within the spaceship it would be a way to tell your new 0.75c inertial reference frame is different from the one you started at.

And that's why you can't do it - it would violate the principle of relativity. No such violation has ever been detected.

What if you looked out a window and noted the color of the starlight was doppler shifted? Could you measure your speed that way?

You could measure your speed relative to the stars this way, yes. It's how astronomers measure the speeds of stars as a matter of routine.

Are we totally sure this is not an exception to the no privileged referenced frame thing? It's not the slowing down itself that bothers me. It's the apparent violation of the all inertial reference frames are equally valid rule.

There is no violation here. If we both start accelerating at 1g in opposite directions and monitor our own acceleration and the acceleration of the other, we will both say we accelerate constantly at 1g and the other accelerates at an ever decreasing rate. If we both apply our knowledge of special relativity, we will both be able to determine that the other feels a so-called "proper acceleration" (proper in the Latin sense of property, our own acceleration, not in the modern English sense of "real") that is constant.

Does anyone know of any books that go into depth exploring this particular aspect of SR?

See previous post.

 

[A.T.]

It's the apparent violation of the all inertial reference frames are equally valid rule.

It's derived from the two postulates, one of which is the equivalence of inertial reference frames. So it's not a violation of it.

 

[Mister T]

The ship's acceleration would get slower and slower until it was basically stuck at a certain speed relative to the earth.

Not stuck, just very very close to speed $c$, and getting closer all the time even though the increase may become so small that your measuring devices can't detect it. Note that this happens (with particles instead of ships) every day in particle accelerators all over the world.

So it's a bit like some magical force, the hand of a god, is holding them back from increasing their speed despite the 1g of acceleration.

Only to someone who thinks that Newtonian physics is valid at low speeds and that special relativity is valid only at high speeds.

Or in other words, $F=ma$ at low speeds and $F=\gamma^3 ma$ at high speeds.

The correct way to think about it is that $F=\gamma^3 ma$ for all speeds and that $\gamma^3 \approx 1$ at low speeds, so close to $1$ that your measuring devices indicate it's equal to $1$.

 

[russ_watters]

Are we totally sure this is not an exception to the no privileged referenced frame thing? It's not the slowing down itself that bothers me. It's the apparent violation of the all inertial reference frames are equally valid rule. The increase in (unmeasurable from the local frame) apparent mass only happens when the reference frame is traveling at close to c with respect to something. Possibly with respect to anything in the universe.
[separate post]
So it's a bit like some magical force, the hand of a god, is holding them back from increasing their speed despite the 1g of acceleration.

That's probably why the concept of "relativistic mass" isn't used anymore; it doesn't make sense. The wiki article on the subject includes this quote:

"...it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself."

https://en.m.wikipedia.org/wiki/Mass_in_special_relativity